Solving 5x² - 2x + 5 = 0 Using The Quadratic Formula

In this comprehensive exploration, we will meticulously solve the quadratic equation 5x² - 2x + 5 = 0. To effectively tackle this problem, we will leverage the quadratic formula, a cornerstone in solving equations of this form. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form ax² + bx + c = 0, where a, b, and c are coefficients, and x represents the variable we aim to find. The quadratic formula provides a direct method to determine the solutions (also known as roots) of such equations. It is given by:

$$x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}$$

This formula is derived by completing the square on the general form of the quadratic equation and is a fundamental tool in algebra. Understanding and applying this formula correctly is crucial for solving a wide range of mathematical problems. In our specific case, the equation 5x² - 2x + 5 = 0 fits this standard form, allowing us to identify the coefficients and apply the formula systematically. The beauty of the quadratic formula lies in its universality; it works regardless of whether the roots are real or complex, rational or irrational. This makes it an indispensable tool for anyone working with quadratic equations. Before we dive into the calculations, it’s essential to grasp the significance of each term in the formula and how they correspond to the coefficients in the equation. The term under the square root, b² - 4ac, known as the discriminant, plays a crucial role in determining the nature of the roots. It tells us whether the roots are real and distinct, real and equal, or complex conjugates. Complex roots arise when the discriminant is negative, indicating that the solutions involve the imaginary unit i, where i² = -1. With this foundational understanding, we are well-equipped to tackle the equation 5x² - 2x + 5 = 0 and unravel its solutions.

To embark on solving the quadratic equation 5x² - 2x + 5 = 0, our initial step involves precisely identifying the coefficients a, b, and c from the standard form equation ax² + bx + c = 0. A meticulous identification process ensures we apply the quadratic formula with the correct values, which is crucial for arriving at the accurate solutions. In our given equation, the coefficient of the x² term, a, is 5. This value represents the quadratic term's contribution to the equation's overall behavior. The coefficient of the x term, b, is -2. Note the inclusion of the negative sign, which is often a point of error if overlooked. This linear term significantly influences the equation's properties and the position of its graph. Lastly, the constant term, c, is 5. This constant value shifts the parabola vertically and is vital in determining the roots of the equation. With the coefficients clearly identified as a = 5, b = -2, and c = 5, we are now primed to substitute these values into the quadratic formula:

$$x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}$$

This formula serves as our primary tool for solving quadratic equations, and its accurate application hinges on the correct identification of coefficients. Substituting the values, we get:

$$x = \frac{-(-2) \pm \sqrt{(-2)² - 4(5)(5)}}{2(5)}$$

The next phase involves simplifying this expression, meticulously following the order of operations to ensure accuracy. The process of simplification is where potential errors can creep in, so a careful step-by-step approach is paramount. We begin by addressing the negative signs and the calculations within the square root. This substitution is a crucial juncture in the solving process, and the accuracy here will directly impact the final result. We are now set to simplify the expression further and uncover the roots of the equation.

Following the substitution of coefficients into the quadratic formula, our next critical step is to simplify the resulting expression. This process begins with evaluating the terms inside the square root, also known as the discriminant. Recall our expression:

$$x = \frac{-(-2) \pm \sqrt{(-2)² - 4(5)(5)}}{2(5)}$$

First, let's simplify the terms. We have -(-2) which simplifies to 2. Next, we calculate the value under the square root. (-2)² equals 4, and 4(5)(5) equals 100. Thus, the expression under the square root becomes 4 - 100, which equals -96. The discriminant, therefore, is -96. This negative value is particularly significant because it indicates that the solutions to the quadratic equation will be complex numbers. Complex numbers involve the imaginary unit i, where i² = -1. The presence of a negative discriminant underscores that the parabola represented by the quadratic equation does not intersect the x-axis, hence the absence of real roots. Now, let's continue simplifying the denominator. 2(5) equals 10. Substituting these simplified values back into the equation, we now have:

$$x = \frac{2 \pm \sqrt{-96}}{10}$$

The next challenge is to handle the square root of -96. To do this, we will express the square root of -96 in terms of the imaginary unit i and simplify it. This involves factoring out the perfect square factors from 96 and expressing the negative sign using i. Simplifying the square root correctly is crucial for obtaining the final solutions in their simplest form. The presence of the imaginary unit i signals that we are dealing with complex roots, which are pairs of numbers in the form a + bi and a - bi, where a and b are real numbers. With the discriminant calculated and the initial simplifications complete, we are well-positioned to further simplify the expression and extract the complex roots of the equation.

Having arrived at the expression x = (2 ± √(-96)) / 10, our primary focus now shifts to simplifying the square root of -96. To achieve this, we will express √(-96) in terms of the imaginary unit i and then further simplify the radical. Remember that i is defined as √(-1), which allows us to rewrite √(-96) as √(96) * √(-1) or √(96) * i. The next step is to find the prime factorization of 96 to identify any perfect square factors that can be extracted from the square root. The prime factorization of 96 is 2 × 2 × 2 × 2 × 2 × 3, which can be grouped as (2² × 2² × 2 × 3). This allows us to rewrite √(96) as √(2² × 2² × 2 × 3), which simplifies to 2 × 2 × √(2 × 3) or 4√6. Therefore, √(-96) becomes 4i√6. Substituting this back into our equation, we have:

$$x = \frac{2 \pm 4i\sqrt{6}}{10}$$

Now, we can see that all the terms in the numerator and the denominator share a common factor of 2. Dividing each term by 2, we simplify the fraction to its simplest form. This step is crucial for presenting the solution in its most concise and understandable form. Simplifying fractions is a fundamental algebraic skill, and here it helps us to clearly reveal the structure of the complex roots. After dividing by 2, we obtain:

$$x = \frac{1 \pm 2i\sqrt{6}}{5}$$

This is the simplified form of the solution, expressing the roots of the quadratic equation as complex numbers. The roots are in the form a + bi and a - bi, where a = 1/5 and b = 2√6 / 5. These complex roots are conjugates of each other, a common characteristic of quadratic equations with negative discriminants. We have now successfully simplified the expression and expressed the solution in its standard complex form, making it easy to identify the real and imaginary parts of the roots.

After a meticulous step-by-step simplification process, we have arrived at the final solutions for the quadratic equation 5x² - 2x + 5 = 0. The simplified expression is:

$$x = \frac{1 \pm 2i\sqrt{6}}{5}$$

This result indicates that the quadratic equation has two complex conjugate roots. These roots are:

$$x_1 = \frac{1 + 2i\sqrt{6}}{5}$$

$$x_2 = \frac{1 - 2i\sqrt{6}}{5}$$

These complex roots consist of a real part (1/5) and an imaginary part (±2√6 / 5)i. The presence of complex roots is a direct consequence of the negative discriminant we calculated earlier, which was -96. A negative discriminant is a clear indicator that the quadratic equation has no real roots; instead, its roots lie in the complex number plane. In summary, we methodically applied the quadratic formula, carefully substituted the coefficients, simplified the expression, handled the square root of a negative number by introducing the imaginary unit i, and finally, expressed the solutions in their simplest form. The solution x = (1 ± 2i√6) / 5 precisely captures the roots of the given quadratic equation. This comprehensive process demonstrates the power and versatility of the quadratic formula in solving quadratic equations, even when the roots are complex. Understanding and mastering this process is fundamental for anyone studying algebra and related mathematical fields. This journey through solving 5x² - 2x + 5 = 0 provides a clear example of how complex numbers arise in quadratic equations and how to handle them with precision.

Final Answer: The final answer is (A).